Bayes’ theorem written a certain way is surprisingly effective and easy to use in Fermi estimates of population parameters and risks. Unless you are already quite well versed in intuitive Bayes, this is likely of interest.
Hastie & Dawes (2010, p. 108) describe the “Ratio Rule”, a helpful way of writing out Bayes’ theorem that is useful for the quick estimation of an unknown proportion:
Pr(A|B) / Pr(B|A) = Pr(A) / Pr(B)
(Ratio of conditional probabilities equals ratio of unconditional probabilities.)
To steal their example, it’s often reported that most ‘hard’ drug users also use (or started out with) pot, and this is often taken to support the notion that pot is a gateway to hard drugs. Hastie & Dawes point out that for the purposes of evaluating the ‘gateway’ claim, what we really want is not the reported value of Pr(has used pot | has used hard drugs) but rather Pr(has used hard drugs | has used pot) [*]. Suppose that Pr(pot|hard) ~ 0.9. We know that Pr(pot) ~ 0.5 (fraction of Americans who’ve used pot at some point), and we estimate that Pr(hard) is lower by a factor of, say 2.5 − 5, so the ratio Pr(pot) / Pr(hard) is between 2.5 and 5. By the ratio rule, Pr(pot|hard)/Pr(hard|pot) = 2.5 − 5, so Pr(hard|pot) is between about 0.2 and 0.4.
Another example: recently I found that the annual risk of dying by suicide for a young to middle-aged male is about 0.02%, as high as the annual risk of a middle-aged male dying in a car accident (!). I figured that taking into account my not having a history of mental illness should decrease this risk. Googling revealed that Pr(mental illness|suicide) = 0.9, and Pr(mental illness) is between 0.06 and 0.25 depending on the severity criteria you use. I want x = Pr(suicide|not mental illness), so I set up the ratios (assuming that the population proportions can be interpreted as annual risk):
Bayes’ theorem written a certain way is surprisingly effective and easy to use in Fermi estimates of population parameters and risks. Unless you are already quite well versed in intuitive Bayes, this is likely of interest.
Hastie & Dawes (2010, p. 108) describe the “Ratio Rule”, a helpful way of writing out Bayes’ theorem that is useful for the quick estimation of an unknown proportion:
Pr(A|B) / Pr(B|A) = Pr(A) / Pr(B)
(Ratio of conditional probabilities equals ratio of unconditional probabilities.)
To steal their example, it’s often reported that most ‘hard’ drug users also use (or started out with) pot, and this is often taken to support the notion that pot is a gateway to hard drugs. Hastie & Dawes point out that for the purposes of evaluating the ‘gateway’ claim, what we really want is not the reported value of Pr(has used pot | has used hard drugs) but rather Pr(has used hard drugs | has used pot) [*]. Suppose that Pr(pot|hard) ~ 0.9. We know that Pr(pot) ~ 0.5 (fraction of Americans who’ve used pot at some point), and we estimate that Pr(hard) is lower by a factor of, say 2.5 − 5, so the ratio Pr(pot) / Pr(hard) is between 2.5 and 5. By the ratio rule, Pr(pot|hard)/Pr(hard|pot) = 2.5 − 5, so Pr(hard|pot) is between about 0.2 and 0.4.
Another example: recently I found that the annual risk of dying by suicide for a young to middle-aged male is about 0.02%, as high as the annual risk of a middle-aged male dying in a car accident (!). I figured that taking into account my not having a history of mental illness should decrease this risk. Googling revealed that Pr(mental illness|suicide) = 0.9, and Pr(mental illness) is between 0.06 and 0.25 depending on the severity criteria you use. I want x = Pr(suicide|not mental illness), so I set up the ratios (assuming that the population proportions can be interpreted as annual risk):
Pr(suicide|not mental illness) / Pr(not mental illness|suicide) = Pr(suicide) / Pr(not mental illness)
x/0.1 = 0.0002/0.75 ~ 0.0002
x ~ 0.00002 (0.002%)
This seems much less worth worrying about. In the plausible range, the precise value of Pr(not mental illness) doesn’t matter much.
[*] What we would really like is Pr(has used hard drugs | started out with pot) but we can assume that the two are close.